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https://github.com/quantumjim/Quantum-Computation-course-Basel.git
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75 lines
2.5 KiB
Plaintext
75 lines
2.5 KiB
Plaintext
{
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"# Exercise 3\n",
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"\n",
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"This sheet is about [density matrices](../QI_course/2_The_Qubit.pdf), [the partial trace](../QI_course/3_Quantum_Information.pdf) and [the Schmidt decomposition](../QI_course/6_Quantum_Correlations_part_1.pdf). Take a look at the linked materials to find out more."
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},
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{
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"cell_type": "markdown",
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"## 1\n",
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"\n",
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"Given an arbitrary single qubit state $a|0\\rangle +b|1\\rangle$:\n",
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"\n",
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"(a) Write down the corresponding density matrix.\n",
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"\n",
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"(b) Write down the density matrix representing the effect of applying an $X$ with probability $q_x$, $Y$ with probability $q_y$, $Z$ with probability $q_z$ (and doing nothing with probability $1-q_x-q_y-q_z$).\n",
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"\n",
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"(c) Write down the density matrix for representing the effect of replacing the state with $I/2$ with probability $p$.\n",
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"\n",
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"(d) For what values of $q_x$, $q_y$ and $q_z$ is the effect of (b) equivalent to that of (c)?\n",
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"\n",
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"## 2\n",
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"\n",
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"(a) Write down the density matrix of a single qubit which is in state $|0\\rangle$ with probability $2/3$ and $|+\\rangle$ with probability $1/3$.\n",
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"\n",
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"(b) Find a two qubit state such that one of the reduced density matrices is equal to the density matrix in (a).\n",
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"\n",
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"(c) Write down a density matrix for a state that is not entangled, but which has the same reduced density matrices as your answer for (b).\n",
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"\n",
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"## 3\n",
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"\n",
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"Consider the three qubit state $(|001\\rangle + |010\\rangle)+ |100\\rangle/\\sqrt{3}$.\n",
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"\n",
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"(a) What is the reduced density matrix for a single qubit in this state?\n",
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"\n",
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"(b) What is the reduced density matrix for a pair of qubits in this state?\n",
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"\n",
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"(c) Use the Schmidt decomposition to rewrite the state, with one qubit on one side of the bipartition and two qubits on the other."
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